On a Class of Gibbs Sampling over Networks

Bo Yuan, Jiaojiao Fan, Jiaming Liang, Andre Wibisono, Yongxin Chen
Proceedings of Thirty Sixth Conference on Learning Theory, PMLR 195:5754-5780, 2023.

Abstract

We consider the sampling problem from a composite distribution whose potential (negative log density) is $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\nicefrac{\sigma_{ij}}{2\eta} \Vert x_i-y_j \Vert^2_2$ where each of $x_i$ and $y_j$ is in $\Rd$, $f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ are strongly convex functions, and $\{\sigma_{ij}\}$ encodes a network structure. Building on the Gibbs sampling method, we develop an efficient sampling framework for this problem when the network is a bipartite graph. More importantly, we establish a non-asymptotic linear convergence rate for it. This work extends earlier works that involve only a graph with two nodes \cite{lee2021structured}. To the best of our knowledge, our result represents the first non-asymptotic analysis of a Gibbs sampler for structured log-concave distributions over networks.Our framework can be potentially used to sample from the distribution $ \propto \exp[-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x)]$ in a distributed manner.

Cite this Paper

BibTeX
@InProceedings{pmlr-v195-yuan23a, title = {On a Class of Gibbs Sampling over Networks}, author = {Yuan, Bo and Fan, Jiaojiao and Liang, Jiaming and Wibisono, Andre and Chen, Yongxin}, booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory}, pages = {5754--5780}, year = {2023}, editor = {Neu, Gergely and Rosasco, Lorenzo}, volume = {195}, series = {Proceedings of Machine Learning Research}, month = {12--15 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v195/yuan23a/yuan23a.pdf}, url = {https://proceedings.mlr.press/v195/yuan23a.html}, abstract = {We consider the sampling problem from a composite distribution whose potential (negative log density) is $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\nicefrac{\sigma_{ij}}{2\eta} \Vert x_i-y_j \Vert^2_2$ where each of $x_i$ and $y_j$ is in $\Rd$, $f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ are strongly convex functions, and $\{\sigma_{ij}\}$ encodes a network structure. Building on the Gibbs sampling method, we develop an efficient sampling framework for this problem when the network is a bipartite graph. More importantly, we establish a non-asymptotic linear convergence rate for it. This work extends earlier works that involve only a graph with two nodes \cite{lee2021structured}. To the best of our knowledge, our result represents the first non-asymptotic analysis of a Gibbs sampler for structured log-concave distributions over networks.Our framework can be potentially used to sample from the distribution $ \propto \exp[-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x)]$ in a distributed manner. } }
Endnote
%0 Conference Paper %T On a Class of Gibbs Sampling over Networks %A Bo Yuan %A Jiaojiao Fan %A Jiaming Liang %A Andre Wibisono %A Yongxin Chen %B Proceedings of Thirty Sixth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Gergely Neu %E Lorenzo Rosasco %F pmlr-v195-yuan23a %I PMLR %P 5754--5780 %U https://proceedings.mlr.press/v195/yuan23a.html %V 195 %X We consider the sampling problem from a composite distribution whose potential (negative log density) is $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\nicefrac{\sigma_{ij}}{2\eta} \Vert x_i-y_j \Vert^2_2$ where each of $x_i$ and $y_j$ is in $\Rd$, $f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ are strongly convex functions, and $\{\sigma_{ij}\}$ encodes a network structure. Building on the Gibbs sampling method, we develop an efficient sampling framework for this problem when the network is a bipartite graph. More importantly, we establish a non-asymptotic linear convergence rate for it. This work extends earlier works that involve only a graph with two nodes \cite{lee2021structured}. To the best of our knowledge, our result represents the first non-asymptotic analysis of a Gibbs sampler for structured log-concave distributions over networks.Our framework can be potentially used to sample from the distribution $ \propto \exp[-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x)]$ in a distributed manner.
APA
Yuan, B., Fan, J., Liang, J., Wibisono, A. & Chen, Y.. (2023). On a Class of Gibbs Sampling over Networks. Proceedings of Thirty Sixth Conference on Learning Theory, in Proceedings of Machine Learning Research 195:5754-5780 Available from https://proceedings.mlr.press/v195/yuan23a.html.

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